An analytical-numerical approach to vibration analysis of periodic Timoshenko beams

Domagalski, Łukasz; Świątek, Michał; Jędrysiak, Jarosław

doi:10.1016/j.compstruct.2018.12.007

Cited by 20 publications

(12 citation statements)

References 30 publications

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“…It appears that the considered problems will require further analysis, including when it comes to comparing the results obtained numerically or from experimental studies. Some validations of the tolerance model were done for dynamic problems using the finite element method, which was shown, among others in Reference [ 78 ] for thin functionally graded plates, and in Reference [ 72 ] for periodic Timoshenko beams. The analysis of additional (higher) critical force obtained under the tolerance model also remains open.…”

Section: Conclusion and Remarksmentioning

confidence: 99%

“…This method makes it possible to investigate various dynamical, stability, and thermoelastic problems for periodic structures, which can be found in a series of articles. For instance: fluid-saturated periodic grounds were investigated in Reference [ 61 ]; the dynamics of periodic plane structures was considered in Reference [ 62 ]; the vibrations of wavy-type periodic plates were analysed in Reference [ 63 ]; the dynamics of periodic thin plates reinforced by stiffeners was described in Reference [ 64 ]; an application for vibrations of medium thickness periodic plates was shown in Reference [ 65 ]; a stability of periodic thin plates on a foundation was considered in Reference [ 66 ]; the dynamics of thin periodic plates with the microstructure size of an order of the thickness of the plate was analysed in Reference [ 67 ]; applications for dynamics and stability of shells with a periodic microstructure were presented in References [ 68 , 69 ]; the nonlinear dynamics of periodic visco-elastic plates was shown in Reference [ 70 ]; the geometrically nonlinear dynamics of periodic beams was analysed in Reference [ 71 , 72 ]; the vibrations of three-layered periodic plates were considered in Reference [ 73 ]; an analysis of free vibrations for thin periodic plates having uncertain material properties was presented in Reference [ 74 ].…”

Section: Introductionmentioning

confidence: 99%

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Theoretical Analysis of Buckling for Functionally Graded Thin Plates with Microstructure Resting on an Elastic Foundation

Jędrysiak

Kaźmierczak-Sobińska

2020

Materials

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In this paper, the problem of the stability of functionally graded thin plates with a microstructure is presented. To analyse this problem and take into consideration the effect of microstructure, tolerance modelling is used. The tolerance averaging technique allows us to replace the equation with non-continuous, tolerance-periodic, highly oscillating coefficients of the system of differential equations with slowly-varying coefficients, which describes also the effect of the microstructure. As an example, the buckling of a microstructured functionally graded plate band on a foundation is investigated. To obtain results, the tolerance model and the asymptotic model combined together with the Ritz method are used. It is shown that the tolerance model allows us to take into account the effect of microstructure on critical forces.

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Section: Conclusion and Remarksmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Theoretical Analysis of Buckling for Functionally Graded Thin Plates with Microstructure Resting on an Elastic Foundation

Jędrysiak

Kaźmierczak-Sobińska

2020

Materials

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“…Tomczyk and Szczerba [15], Tomczyk et al [16]), dynamics of beams (cf. Domagalski [17], Domagalski et al [18], Domagalski and Jędrysiak [19]), statics of plates with a dense system of ribs (cf. Marczak et al [20]), dynamics of sandwich plates (cf.…”

Section: Introductionmentioning

confidence: 99%

The Stability Analysis of Periodic Beams Interacting with Periodic Elastic Foundation with the Use of the Tolerance Averaging Technique

Marczak

Jędrysiak

2021

Materials

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In this paper a stability analysis of microperiodic beams resting on the periodic inhomogeneous foundation is carried out. The main issue of this considerations, which is the analytical solution to the governing equations characterised by periodic, highly oscillating and non-continuous coefficients, is overwhelmed by the application of the tolerance averaging technique. As a result of such application, the governing equation is transformed into a form with constant coefficients which can be solved using well-known mathematical methods. In several calculation examples, the convergence of the results of two derived averaged models is examined, as well as the convergence of the lowest value of the critical force parameter derived from the averaged models with the FEM model. The results prove the superiority of the presented analytical solution over the FEM analysis in the optimisation process.

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“…However, only a few studies have addressed the nonlinear dynamics of finite-size periodic beams. An asymptotic tolerance averaging technique was applied to the nonlinear vibration analysis of a beam with geometric nonlinearity and periodic distribution parameters, but this study was limited to low-frequency vibration and a high parameter-varying wave because of the averaging in a periodic unit [59,60]. Those analysis methods have limitations in the number of used vibration modes which can affect the accuracy and reliability of results as the nonlinearity and periodicity will induce the coupling of linear structural modes.…”

Section: Introductionmentioning

confidence: 99%

Vibration Localization and Anti-Localization of Nonlinear Multi-Support Beams with Support Periodicity Defect

Ying

2021

Symmetry

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A response analysis method for nonlinear beams with spatial distribution parameters and non-periodic supports was developed. The proposed method is implemented in four steps: first, the nonlinear partial differential equation of the beams is transformed into linear partial differential equations with space-varying parameters by using a perturbation method; second, the space-varying parameters are separated into a periodic part and a non-periodic part describing the periodicity defect, and the linear partial differential equations are separated into equations for the periodic and non-periodic parts; third, the equations are converted into ordinary differential equations with multiple modes coupling by using the Galerkin method; fourth, the equations are solved by using a harmonic balance method to obtain vibration responses, which are used to discover dynamic characteristics including the amplitude–frequency relation and spatial mode. The proposed method considers multiple vibration modes in the response analysis of nonlinear non-periodic structures and accounts for mode-coupling effects resulting from structural nonlinearity and parametric non-periodicity. Thus, it can handle nonlinear non-periodic structures with a high parameter-varying wave in wide frequency vibration. In numerical studies, a nonlinear beam with non-periodic supports (resulting in non-periodic distribution parameters or periodicity defect) under harmonic excitations was explored using the proposed method, which revealed some new dynamic response characteristics of this kind of structure and the influences of non-periodic parameters. The characteristics include remarkable variation in frequency response and spatial mode, and in particular, vibration localization and anti-localization. The results have potential applications in vibration control and the support damage detection of nonlinear structures with non-periodic supports.

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An analytical-numerical approach to vibration analysis of periodic Timoshenko beams

Cited by 20 publications

References 30 publications

Theoretical Analysis of Buckling for Functionally Graded Thin Plates with Microstructure Resting on an Elastic Foundation

Theoretical Analysis of Buckling for Functionally Graded Thin Plates with Microstructure Resting on an Elastic Foundation

The Stability Analysis of Periodic Beams Interacting with Periodic Elastic Foundation with the Use of the Tolerance Averaging Technique

Vibration Localization and Anti-Localization of Nonlinear Multi-Support Beams with Support Periodicity Defect

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